On Wednesday, March 21, 2018 at 7:01:43 AM UTC-5, Lawrence Crowell wrote:   
   > I will get the question out of the way. Why is this presuming the   
   > universe is a non-Riemannian manifold or geometry? General relativity   
   > is physics based on Riemannian differential geometry.   
      
   Actually, General Relativity is based primarily on [in order of   
   importance] (1) the equivalence principle, (2) the continuity equations   
   that embody conservation laws for mass, energy and momentum (and angular   
   momentum and moment) and (3) the assumption that the geometry is locally   
   3+1 -- conservation laws that (as Lydia pointed out) are abruptly   
   violated on any "singularities" in a solution, thereby undermining one   
   of the chief premises used to establish the theory! For this reason, any   
   solution with a singularity has to be excluded from   
   consideration. Therefore, a no-go theorem that essentially makes   
   singularities inevitable can only be taken as a proof by contradiction   
   that the basic assumption (of the geometry being Riemannian) is false.   
      
   Also, on a technical note, the theory can only be founded properly on a   
   Riemann-*Cartan* geometry, since Riemannian geometries do not support   
   the infrastructure required to embody non-natural objects (such as a   
   Lorentz bundle or SL(2,C) bundle -- which you need even to express the   
   idea of a spinor, never mind expressing anything involving them!) You   
   need also the 3-currents for angular momentum and moment.   
      
   So, (1), (2) and (3) together mandate a Riemann-Cartan geometry, not a   
   Riemannian geometry; and General Relativity (where but for the   
   historical accident of having predated the formulation of Riemann-Cartan   
   geometry by about 10 years) is actually founded on a Riemann-Cartan   
   geometry. The only real choice is not between geometry types, then, but   
   whether to adopt the Einstein-Hilbert Lagrangian (which, when rewritten   
   in intrinsic form in a Riemannian-Cartan geometry is very awkward, since   
   it requires adding extra terms to subtract out the contorsion from the   
   native connection so as to get the Levi-Civita connection) or the   
   Riemann-Cartan "native" form of the Einstein-Hilbert Lagrangian (which   
   uses the Riemann-Cartan's native R scalar, rather than the R based on   
   the Levi-Civita connection). That gives you one of 2 theories:   
   Einstein-Hilbert(-in-Riemann-Cartan-geometry-form) or   
   Einstein-Cartan. The empirical differences between the two are very   
   small (but could affect early cosmology, as Trautman has pointed out!),   
   so the matter is not yet decided. Exterior solutions will not see any   
   difference, since both torsion and contorsion cannot propagate outside   
   matter in Einstein-Cartan gravity (nor in any of a large range of other   
   Lagrangians built on a Riemann-Cartan geometry) ... by a simple counting   
   argument on number of degrees of freedom for the spin tensor and torsion   
   tensor.   
      
   Note that assumptions (1) and (2) do not mandate a Riemannian geometry   
   nor one that is pseudo-Riemanninan. For instance, Newton's law of   
   gravity falls within this mould but is described with a (most decidedly   
   *non*-[pseudo-]Riemannian) Newton-Cartan geometry. The only qualifier to   
   that is that it can be embedded in a 4+1 geometry as I, myself, have   
   pointed out here on a previous occasion, such that the geodesic law   
   directly embodies the Newtonian law of gravity. In fact, you can easily   
   find such a metric by taking the Schwarzschild metric and substituting   
   the Lydia's invariant (ds = dt + v du) in for the proper time and 1/c^2   
   for v, and make it applicable to all motion under the influence of a   
   potential G(r) by substituting -GM/r by G(r). Move all the terms for the   
   proper time over to the same side as the line element, cancel out the   
   dt^2 terms, and rescale so that the metric has the asymptotic form   
   dx^2 + dy^2 + dz^2 + ... = 0 (as v -> 0). Note what extra terms crop up in   
   the 4+1 metric where G is present.   
      
   > The CMB occurred relatively late in the evolution of the universe,   
   > here late being compared to inflationary cosmology etc.   
      
   Actually, the CMB occurred at the *end* of the radiation dominant   
   phase. The radiation dominant phase existed (as far as anyone can   
   determine) at all earlier times before that.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)